A private conversation between a customer on my Facebook business page didn't create a ticket. An individual reply from the customer or a Facebook page administrator, outside of Support, didn't create a ticket.
The are several reasons for a private message to not appear in Zendesk.
If the messages for certain conversations don't create tickets in Support but others work as expected, the user likely disabled access by third-party apps as discussed in this article.
Individual replies missing
If individual replies are not showing, the conversation was likely archived on Facebook.
Social Messaging Add-on
With the Social Messaging add-on, you can send and receive private messages from Facebook Messenger. Without this add-on, private messages from Facebook do not create tickets within your Support account. For more information, see the article: Installing and setting up the Social Messaging add-on.
To continue conversations with users who have disabled access by third-party apps, have a Facebook page administrator replying to the user directly on Facebook. If the user changes their Facebook permissions in the future, messages after that point should pull in successfully.
Encourage Facebook page administrators to not archive conversations so quickly. The replies should pull over in a matter of minutes, but archiving a conversation immediately after a reply can cause the comments to not arrive in the Support ticket.